Linear Spaces

Convergence

Definition: Let (V,F)(V,F) be a vector space. A sequence of vectors {xn}n=1∞\{x_n\}_{n=1}^{\infty} in VV is said to converge to a vector x∈Vx \in V if for every ϵ>0\epsilon > 0 there exists an integer NN such that ∥xn−x∥<ϵ\|x_n - x\| < \epsilon for all n≥Nn \geq N. In this case we write xn→xx_n \rightarrow x as n→∞n \rightarrow \infty.

Remark: The sequence is said to be convergent if it converges to some vector x∈Vx \in V. Otherwise, it is said to be divergent.

Example: Let V=R  ∥v∥=vV=R \ \ \|v\|=v, consider the sequence

{(12)n}n=1∞\left\{ (\frac{1}{2})^n \right\}_{n=1}^{\infty}

Proove that this sequence converges to 00 as n→∞n \rightarrow \infty.

 ~Proof: DO IT LATER

Definition: Let (V,F,∥.∥)(V,F, \|.\|) be a normed space. A sequence of vectors {xn}n=1∞\{x_n\}_{n=1}^{\infty} in VV is said to be a Cauchy sequence if for every ϵ>0\epsilon > 0 there exists an integer NN such that ∥xn−xm∥<ϵ\|x_n - x_m\| < \epsilon for all n,m≥Nn,m \geq N.


#EE501 - Linear Systems Theory at METU